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href="/nlab/show/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#stable_homotopy_elements'>Stable homotopy elements</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#IntersectionPairing'>Relation to intersection pairing and Kervaire invariant</a></li> </ul> <li><a href='#properties_2'>Properties</a></li> <ul> <li><a href='#moduli_of_framings'>Moduli of framings</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#for_2manifolds'>For 2-manifolds</a></li> <li><a href='#for_3manifolds'>For 3-manifolds</a></li> <li><a href='#left_invariant_framings_on_compact_connected_lie_groups'>Left invariant framings on compact connected Lie groups</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>In one sense of the term, a <em>framing</em> of a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> is a choice of trivialization of its <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, hence a choice of <a class="existingWikiWord" href="/nlab/show/section">section</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a>.</p> <p>A manifold that admits a framing is also called a <strong>parallelizable manifold</strong>. A manifold equipped with a framing is also called a <strong>parallelized manifold</strong>.</p> <p>More generally, one means by a <em>framing</em> not a trivialization of the tangent bundle itself, but</p> <ul> <li> <p>of the <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a> if the manifold is understood <a class="existingWikiWord" href="/nlab/show/embedding">embedded</a> in some <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math></p> </li> <li> <p>of the <em><a class="existingWikiWord" href="/nlab/show/stable+tangent+bundle">stable tangent bundle</a></em>.</p> </li> </ul> <p>Accordingly, a <em>framed cobordism</em> is a <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> equipped with a framing on the underlying manifold (see also at <a class="existingWikiWord" href="/nlab/show/MFr">MFr</a>).</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">dim(X)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of the manifold and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n \geq dim(X)</annotation></semantics></math>, then one also speaks of an <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-framing</em> to mean a trivialization of the “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-stabilized tangent bundle” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>⊕</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>−</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">T X \oplus \mathbb{R}^{n-dim(X)}</annotation></semantics></math> (where the right <a class="existingWikiWord" href="/nlab/show/direct+sum">direct summand</a> denotes the trivial real <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> of <a class="existingWikiWord" href="/nlab/show/rank">rank</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n - dim(X)</annotation></semantics></math>). These <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-framed manifolds appear in particular in the construction of the <a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a> of framed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional cobordisms. The <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> asserts essentially that the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-framing is the <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2Cn%29-category+with+duals">symmetric monoidal (∞,n)-category with duals</a>.</p> <p>Beware that there is also the term “<a class="existingWikiWord" href="/nlab/show/2-framing">2-framing</a>” due to (<a href="#Atiyah">Atiyah</a>), which is related but different.</p> <h2 id="examples">Examples</h2> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> is a parallelizable manifold.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Every non-zero <a class="existingWikiWord" href="/nlab/show/invariant+vector+field">invariant vector field</a> on the Lie group provides an everywhere non-vanishing section of the tangent bundle.</p> </div> <p>The following is obvious:</p> <div class="num_prop" id="SpinManifoldAdmitsFraming"> <h6 id="proposition_2">Proposition</h6> <p>Every 3-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> with <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> admits a framing.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>That a 3-manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> means that we have a <a class="existingWikiWord" href="/nlab/show/reduction+of+the+structure+group">reduction of the structure group</a> of the tangent bundle to the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>, and hence the tangent bundle is classified by a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>B</mi><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to B Spin(3)</annotation></semantics></math>. But <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3)</annotation></semantics></math> has vanishing <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">0 \leq k \leq 2</annotation></semantics></math>. Therefore its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B Spin(3)</annotation></semantics></math> has vanishing homotopy groups below degree 4, and hence every morphism out of a 3-dimensional manifold into it is homotopically constant.</p> </div> <p>But in fact, the following stronger statement is also true.</p> <div class="num_prop" id="EveryOrientable3ManifoldsIsParallelizable"> <h6 id="proposition_3">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/orientation">orientable</a> 3-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> admits a framing.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By the argument in the proof of prop. <a class="maruku-ref" href="#SpinManifoldAdmitsFraming"></a>, the only possible obstruction is the <a class="existingWikiWord" href="/nlab/show/second+Stiefel-Whitney+class">second Stiefel-Whitney class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">w_2</annotation></semantics></math>. By the discussion at <em><a class="existingWikiWord" href="/nlab/show/Wu+class">Wu class</a></em>, this vanishes on an oriented manifold precisely if the second <a class="existingWikiWord" href="/nlab/show/Wu+class">Wu class</a> vanishes. This, in turn, is by definition defined to represent the <a class="existingWikiWord" href="/nlab/show/Steenrod+square">Steenrod square</a> under <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a>, and this vanishes on a 3-manifold by degree reasons.</p> </div> <div class="num_remark" id="In4dFirstOrderGravity"> <h6 id="remark">Remark</h6> <p>Prop. <a class="maruku-ref" href="#EveryOrientable3ManifoldsIsParallelizable"></a> has some impact in the context of the <a class="existingWikiWord" href="/nlab/show/first+order+formulation+of+gravity">first order formulation of gravity</a>, where one is interested in <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> fields on 4-dimensional <a class="existingWikiWord" href="/nlab/show/spacetime+manifolds">spacetime manifolds</a>, and in particular in <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetimes">globally hyperbolic spacetimes</a>, which, as <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, are the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> (time) with a <a class="existingWikiWord" href="/nlab/show/3-manifold">3-manifold</a> (space). Prop. <a class="maruku-ref" href="#EveryOrientable3ManifoldsIsParallelizable"></a> implies that already when that spatial 3-manifold is orientable, then the whole globally hyperbolic spacetime admits a framing. (See also at <em><a class="existingWikiWord" href="/nlab/show/teleparallel+gravity">teleparallel gravity</a></em>.)</p> </div> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/spheres">spheres</a> that admit a framing are precisely only</p> <ul> <li> <p>the 0-sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mo>=</mo><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">S^0 = \ast \coprod \ast</annotation></semantics></math>, the unit <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></p> </li> <li> <p>the 1-sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> underlying the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> (the unit <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>);</p> </li> <li> <p>the 3-sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math>, underlying the <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math>, is isomorphic to the unit <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a>;</p> </li> <li> <p>the 7-sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math>, which underlies a <a class="existingWikiWord" href="/nlab/show/Moufang+loop">Moufang loop</a> internal to <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>, namely the unit <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a>,</p> </li> </ul> <p>where the algebras appearing are precisely the four <a class="existingWikiWord" href="/nlab/show/normed+division+algebras">normed division algebras</a>.</p> </div> <p>This is due to (<a href="#Adams58">Adams 58</a>), proven with the <a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a>.</p> <h3 id="stable_homotopy_elements">Stable homotopy elements</h3> <p>Left invariant framings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> on compact connected Lie groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">dim(G)=k</annotation></semantics></math> give rise to elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>ℒ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[G,\mathcal{L}]</annotation></semantics></math> in the stable homotopy group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mi>k</mi> <mi>s</mi></msubsup></mrow><annotation encoding="application/x-tex">\pi_k^s</annotation></semantics></math> of spheres. One can restrict attention to semisimple Lie groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> since this construction behaves well with respect to products, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>T</mi><mo>×</mo><mi>G</mi><mo stretchy="false">/</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">G \to T\times G/T</annotation></semantics></math> gives a framed diffeomorphism and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>ℒ</mi><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>π</mi> <mn>1</mn> <mi>s</mi></msubsup></mrow><annotation encoding="application/x-tex">[S^1,\mathcal{L}] \in \pi_1^s</annotation></semantics></math> is the generator. The following facts are assembled from (<a href="#Ossa82">Ossa 1982</a>) and (<a href="#Minami16">Minami 2016</a>).</p> <ul> <li> <p>For any <em>semisimple</em> compact connected Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, Ossa proved that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>72</mn><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>ℒ</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">72[G,\mathcal{L}]=0</annotation></semantics></math>. In particular, the only possible torsion is at the primes 2 and 3.</p> </li> <li> <p>The left invariant framings on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(n),Spin(n)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(n)</annotation></semantics></math> give the zero element for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">n\geq 7</annotation></semantics></math>, those on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(n)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n\geq 4</annotation></semantics></math>, as do those on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>E</mi> <mn>6</mn></msub><mo>,</mo><msub><mi>E</mi> <mn>7</mn></msub><mo>,</mo><msub><mi>E</mi> <mn>8</mn></msub><mo>,</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>,</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mo>,</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>,</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_4,E_6,E_7,E_8, SO(4),SO(6),SU(5),SU(6)</annotation></semantics></math>. (this means that the undetermined cases of rank 4 in Ossa’s Table 1 are, in fact, all 0)</p> </li> <li> <p>The left invariant framings on the remaining groups (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>,</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mo>,</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>,</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>,</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>,</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)=Spin(3)=Sp(1),SU(3),SU(4)=Spin(6),Sp(2)=Spin(5),Sp(3),SO(3),SO(5)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>) represent known nonzero classes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mo>*</mo> <mi>s</mi></msubsup></mrow><annotation encoding="application/x-tex">\pi_\ast^s</annotation></semantics></math>.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="IntersectionPairing">Relation to intersection pairing and Kervaire invariant</h3> <p>On a framed manifold, there is a canonical <a class="existingWikiWord" href="/nlab/show/quadratic+refinement">quadratic refinement</a> of the <a class="existingWikiWord" href="/nlab/show/intersection+pairing">intersection pairing</a>. The associated invariant is the <em><a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></em>.</p> <h2 id="properties_2">Properties</h2> <h3 id="moduli_of_framings">Moduli of framings</h3> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the <a class="existingWikiWord" href="/nlab/show/moduli+space+of+framed+manifolds">moduli space of framings</a> on a fixed manifold is a disjoint union of subgroups of the oriented <a class="existingWikiWord" href="/nlab/show/mapping+class+group">mapping class group</a> which fix a given isotopy type of framings.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+framing">normal framing</a>, <a class="existingWikiWord" href="/nlab/show/normal+twisted+framing">normal twisted framing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/framed+link">framed link</a>, <a class="existingWikiWord" href="/nlab/show/framed+braid">framed braid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/framed+elliptic+curve">framed elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space+of+framed+manifolds">moduli space of framed manifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/teleparallel+gravity">teleparallel gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-framing">2-framing</a></p> </li> </ul> <p>Formalization in <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> is discussed <a href="http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#GLnTangentBundles">there</a>.</p> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Review in the context of the <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a> includes</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>, <em>The Kervaire invariant I</em> (<a href="https://amathew.wordpress.com/2012/10/01/the-kervaire-invariant-i/#more-3888">web</a>)</li> </ul> <p>The theorem about the parallizablitiy of spheres is due to</p> <ul> <li id="Adams58"><a class="existingWikiWord" href="/nlab/show/John+Adams">John Adams</a>, <em>On the Non-Existence of Elements of Hopf Invariant One</em> Bull. Amer. Math. Soc. 64, 279-282, 1958, Ann. Math. 72, 20-104, 1960.</li> </ul> <p>Relation to existence of <a class="existingWikiWord" href="/nlab/show/flat+connections">flat connections</a> on the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> is discussed in</p> <ul> <li id="Thorpe65"><a class="existingWikiWord" href="/nlab/show/John+Thorpe">John Thorpe</a>, <em>Parallelizablility and flat manifolds</em>, 1965 (<a class="existingWikiWord" href="/nlab/files/ThorpeParallelizable.pdf" title="pdf">pdf</a>)</li> </ul> <h3 id="for_2manifolds">For 2-manifolds</h3> <p>The <a class="existingWikiWord" href="/nlab/show/moduli+space+of+framed+surfaces">moduli space of framed surfaces</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Oscar+Randal-Williams">Oscar Randal-Williams</a>, <em>Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces</em> (<a href="http://arxiv.org/abs/1001.5366">arXiv:1001.5366</a>)</li> </ul> <h3 id="for_3manifolds">For 3-manifolds</h3> <p>For <a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Rob+Kirby">Rob Kirby</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Melvin">Paul Melvin</a>: <em>Canonical framings for 3-manifolds</em>, Turkish Journal of Mathematics <strong>23</strong> 1 (1999) [<a class="existingWikiWord" href="/nlab/files/KirbyMelvon3Framings.pdf" title="pdf">pdf</a>]</li> </ul> <p>The relation to “<a class="existingWikiWord" href="/nlab/show/2-framings">2-framings</a>”:</p> <ul> <li id="Atiyah"><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <em>On framings of 3-manifolds</em> (<a href="http://www.maths.ed.ac.uk/~aar/papers/atiyahfr.pdf">pdf</a>)</li> </ul> <h3 id="left_invariant_framings_on_compact_connected_lie_groups">Left invariant framings on compact connected Lie groups</h3> <ul> <li id="Ossa82"> <p>Erich Ossa, Lie groups as framed manifolds, Topology, 21 (1982), 315–323, (<a href="https://doi.org/10.1016/0040-9383%2882%2990013-1">doi</a>)</p> </li> <li id="Minami16"> <p>Haruo Minami, <em>On framed simple Lie groups</em>, (<a href="http://projecteuclid.org/euclid.jmsj/1468956166">pdf</a>)</p> </li> </ul> <p>For an extension to <a class="existingWikiWord" href="/nlab/show/p-compact+groups">p-compact groups</a> see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tilman+Bauer">Tilman Bauer</a>, <em>p-compact groups as framed manifolds</em>, (<a href="https://people.kth.se/~tilmanb/publication/p-compact-groups-as-framed-manifolds/">paper</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 20, 2025 at 11:35:51. 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